Fréchet–Urysohn subspaces of free topological groups

Let F(X)F(X) be the free topological group on a Tychonoff space X. For all natural numbers n   we denote by Fn(X)Fn(X) the subset of F(X)F(X) consisting of all words of reduced length ≤n. In [10], the author found equivalent conditions on a metrizable space X   for F3(X)F3(X) to be Fréchet–Urysohn, and for Fn(X)Fn(X) to be Fréchet–Urysohn for n≥5n≥5. However, no equivalent condition on X   for n=4n=4 was found. In this paper, we prove that for a locally compact metrizable space X such that the set of all non-isolated points of X   is compact, F4(X)F4(X) is Fréchet–Urysohn if F4(X)F4(X) is a k-space. We obtain as a corollary that if a metrizable space X is locally compact separable and the set of all non-isolated points of X   is compact, then F4(X)F4(X) is Fréchet–Urysohn. Consequently, for a metrizable space X=C⊕DX=C⊕D such that C is an infinite compact set and D is a countably infinite discrete subset of X  , F4(X)F4(X) is Fréchet–Urysohn but F5(X)F5(X) is not.